I don't see a way to solve this boot problem...

I have a 22" iMac that is approximately 2 years old. I've never had an issue with it. There is very little 3rd party software and applications installed. On Sunday I was online (Safari) when it froze and I got the rainbow-colored spinning wheel. I waited several minutes for it to unfreeze but it didn't. I reset it from the back and it would not reboot past the gray screen with the apple and spinning loading wheel.

I did my research on this board and tried Safe Boot, resetting PRAM, and repairing the disc via Disc Utility but to no avail. I then attempted an Auto Archive and Install using the Mac OS X discs I received with the computer. I got through the first disc after several attempts and the second disc loaded to completion. The issue is my mouse wouldn't work (batteries are fine) so I couldn't click "Continue" after the second disc loaded. I had to do another reset and now I feel like I'm screwed since when I reset my iMac I can't use any key commands (to perform a safe boot, boot from disc or eject the disc) and it will not load past the same grey screen with the apple and spinning loading wheel.

It sounds like you have a hard drive that is failing. A reinstall almost never solves problems with a bad or failing hard drive.

Also, whenever you reinstall the operating system always make sure you have a wired keyboard and mouse on hand. After the system is up and running you can use blue tooth or whatever.

Is your iMac covered by warranty (Apple care)? If it is, take it on in to Apple and let them diagnose it for you. If the hard drive is indeed bad, they'll change it out. If your machine is not under warranty and you think you can change the hard drive yourself, go to iFixit: The free repair manual and look over their step by step procedure.

Replacing a hard drive in an iMac is a fairly involved procedure and requires the front panel display to be removed first.